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On Convolution Squares of Singular Measures On Convolution Squares of Singular Measures by Vincent Chan A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics in Pure Mathematics Waterloo, Ontario, Canada, 2010 ⃝c Vincent Chan 2010 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Abstract We prove that if 1 > > 1=2, then there exists a probability measure such that the Hausdorff dimension of its support is and ∗ is a Lipschitz function of class − 1=2. v Acknowledgements I would like to deeply thank my supervisor, Kathryn Hare, for her excellent supervision; her comments were useful and insightful, and led not only to finding and fixing mistakes, but also to a more elegant way of presenting various arguments. A careful comparison of earlier drafts of this document and its current version would reveal the subtle errors detected by her keen eye. This thesis would not exist at all if it were not for Professor Thomas K¨orner,whom I thank for his help in clarifying points in his paper, which is the central focus in the pages to come. I am indebted to my friends and colleagues for their support and advice, particularly that of Faisal Al-Faisal, who, being in a similar situation to myself (including the bizarre sleep schedule), has become comparable to a brother in arms. Last but certainly not least, I am grateful for the support of my fianc´eeVicki Nguyen, who, despite having no formal education in pure mathematics, suffered through reading dozens of pages while proofreading. She has never failed to provide encouragement when it was required most, and surely has kept me sane throughout the process. vii Contents 1 Preliminaries1 1.1 Introduction...................................1 1.2 A tightness condition..............................7 2 A Probabilistic Result 11 3 A Key Lemma 21 4 The Main Theorem 35 4.1 -Lipschitz functions.............................. 35 4.2 Density results................................. 41 APPENDICES 57 A Measure Theory 59 B Convergence of Fourier Series 63 C Probability Theory 65 D The Hausdorff Metric 67 Bibliography 74 ix Chapter 1 Preliminaries Unless otherwise stated, we will be working on T = R=Z = [−1=2; 1=2], and only with positive, Borel, regular measures. We will use to denote the Lebesgue measure on T (and dx, dy, dt, and so forth to denote integration with respect to Lebesgue measure), and jEj = (E). Measures that are absolutely continuous or singular are taken with respect to in the case of T, and (left) Haar measure, should we require a more general setting. When speaking of measures living in an Lp space or having other properties associated to functions, we mean the measure is absolutely continuous and we consider its Radon- Nikodym derivative to have this property. 1.1 Introduction The absolutely continuous measures form an ideal in the algebra of measures, so that for an absolutely continuous measure , ∗ is still absolutely continuous and so the Radon- Nikodym theorem gives rise to an associated function f 2 L1 such that ∗ = f. For any -finite measure , Lebesgue's Decomposition theorem decomposes as = ac + s, where ac is absolutely continuous and s is singular. It is thus natural to ask whether a similar situation holds with singular measures, that is, given a singular measure , does there exist a function f 2 L1 with ∗ = f? This is easily seen to be false, for if 1 P P = n anxn is a discrete measure, then ∗ = n;m anamxn+xm is also a discrete measure. To progress further, it may be helpful to appeal to an extension of Lebesgue's Decom- position theorem, which gives for any -finite measure three unique measures ac, cs and d which are absolutely continuous, continuous singular, and discrete, respectively, such that = ac + cs + d. Then we ask if we might recover the nice property of ∗ being absolutely continuous in the case of being a continuous singular measure. This is still not always the case, as there exist continuous singular measures on T where even their nth convolution power is singular for any n; for such an example we will need a preliminary definition. Definition 1.1. Let −1=2 ≤ ak ≤ 1=2 and define the trigonometric polynomials on T N Y k PN (t) = (1 + 2ak cos 3 t): k=1 It is a standard result that PN are probability measures which converge weak-∗ in M(T) to a probability measure with 8 >1 if n = 0; > < M M (n) = Q P kj b j=1 akj if n = j=1 j3 ; j = ±1; > :>0 otherwise: This measure is called the Riesz product measure associated to fakg. Lemma 1.2. If 1 and 2 are Riesz product measures, then so is 1 ∗ 2. Moreover, if i is associated to fak;ig, then 1 ∗ 2 is associated to fak;1ak;2g. Proof. We calculate the Fourier coefficients of the convolution: 8 >1 if n = 0; <> M M k 1 ∗ 2(n) = 1(n)2(n) = Q P j (1.1) \ b b j=1 akj ;1akj ;2 if n = j=1 j3 ; j = ±1; > :>0 otherwise: 2 Then we can define the trigonometric polynomials on T N Y k PN (t) = (1 + 2ak;1ak;2 cos 3 t); k=1 noticing that −1=2 ≤ −1=4 ≤ ak;1ak;2 ≤ 1=4 ≤ 1=2. The weak-∗ limit of PN produces a Riesz product measure whose coefficients satisfy (1.1), so we are finished by uniqueness. A standard result (see, for example, [6] Theorem 7.2.2) gives that a Riesz product P1 2 measure is singular if and only if k=1 ak = 1. In particular, suppose is a Riesz product measure associated to the constant sequence fcg for some constant 0 < c ≤ 1=2. P1 2 n Certainly k=1 c = 1, so that is singular. Inductively applying Lemma 1.2, = ∗ ∗ ⋅ ⋅ ⋅ ∗ (the nth convolution power of ) is a Riesz product measure associated to n P1 2n n fc g. Then k=1 c = 1, and so remains singular. It is also not hard to show that Riesz product measures are continuous, in general. It should be remarked that the choice of frequencies 3k is not essential, and Riesz product measures may be generalized further. In addition, the concept of Riesz product measures as a whole and the conclusion that there exist continuous singular measures whose convolution powers always remain singular can be extended to any infinite, compact, abelian group (see [6]). We now know it is impossible to hope the square convolution of a singular measure is absolutely continuous in general, even when restricting to the case of continuous singular measures. However, we may expect that are still some positive results; convolution behaves a smoothing operation, evidenced by the fact that both the continuous measures and the absolutely continuous measures form ideals in the space of measures. Indeed, a famous result due to Wiener and Wintner [16] produces a singular measure on T such that −1=2+" b(n) = o(jnj ) as n ! 1 for every " > 0. By an application of the Hausdorff-Young inequality, such a measure can be shown to have the property that its square convolution ∗ is absolutely continuous, and in fact, ∗ 2 Lp(T) for every p ≥ 1. Other authors have generalized this result; Hewitt and Zuckerman [9] constructed a singular measure on any non-discrete, locally compact, abelian group G such that ∗ 2 Lp(G) for all p ≥ 1. It is worth noting that the condition of non-discrete cannot be 3 dropped, since otherwise all measures would be absolutely continuous. Using Rademacher- Riesz products, Karanikas and Koumandos [10] prove the existence of a singular measure with ∗ 2 L1(G) for any non-discrete, locally compact group G. Shortly thereafter, Dooley and Gupta [3] produced a measure with ∗ 2 Lp(G) for every p ≥ 1 in the case of compact, connected groups and compact Lie groups using the theory of compact Lie groups. Saeki [15] took this concept further in a different direction by proving the existence of a singular measure on T with support having zero Lebesgue measure such that ∗ has a uniformly convergent Fourier series. This is an improvement on previous results; in this case the continuity of the partial Fourier sums for ∗ ensure that their limit, namely ∗ , is continuous. Then ∗ 2 L1(T) and subsequently ∗ 2 Lp(T) for p ≥ 1 since T is compact. Generalizing this, Gupta and Hare [7] show such a measure exists (now using Haar measure in place of Lebesgue measure) when replacing T with any compact, connected group. K¨orner[11] recently expanded on Saeki's work; to discuss how, we need the following definitions. Definition 1.3. For 1 ≥ ≥ 0, we say a function f : T ! C is Lipschitz of class 1, or simply -Lipschitz and write f 2 Λ if sup sup jhj− jf(t + h) − f(t)j < 1: (1.2) t2T h6=0 Some references define a function to be Lipschitz of class if there is a constant C with jf(x) − f(y)j ≤ Cjx − yj for every x; y 2 T. These definitions are in fact equivalent, with a straightforward proof. Lemma 1.4. f 2 Λ if and only if jf(x) − f(y)j ≤ Cjx − yj for a constant C, for every x; y 2 T. Proof. This is a simple matter of relabeling; use t = y and h = x − y. 1Functions with this property are also sometimes referred to as H¨olderof class .
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